Let ` hat u= hat i+ hat j , hat v= hat i- hat ja n d hat w= hat i+2 hat j+3 hat kdot` If ` hat n` is a unit vector such that ` hat udot hat n=0a n d hat vdot hat n=0,` then find the value of `| hat wdot hat n|dot`

Options

A. `0`
B. `1`
C. `2`
D. `3`

Answer

To solve the problem, we need to find the value of \(|\hat{w} \cdot \hat{n}|\) given that \(\hat{u} \cdot \hat{n} = 0\) and \(\hat{v} \cdot \hat{n} = 0\). Let's break it down step by step.

### Step 1: Define the vectors
We are given:
\[ \hat{u} = \hat{i} + \hat{j} \]
\[ \hat{v} = \hat{i} - \hat{j} \]
\[ \hat{w} = \hat{i} + 2\hat{j} + 3\hat{k} \]

### Step 2: Define the unit vector \(\hat{n}\)
Let \(\hat{n}\) be a unit vector defined as:
\[ \hat{n} = x\hat{i} + y\hat{j} + z\hat{k} \]
Since \(\hat{n}\) is a unit vector, it must satisfy the equation:
\[ x^2 + y^2 + z^2 = 1 \quad \text{(Equation 1)} \]

### Step 3: Use the condition \(\hat{u} \cdot \hat{n} = 0\)
Calculating \(\hat{u} \cdot \hat{n}\):
\[ \hat{u} \cdot \hat{n} = (\hat{i} + \hat{j}) \cdot (x\hat{i} + y\hat{j} + z\hat{k}) = x + y = 0 \]
This gives us:
\[ x + y = 0 \quad \text{(Equation 2)} \]

### Step 4: Use the condition \(\hat{v} \cdot \hat{n} = 0\)
Calculating \(\hat{v} \cdot \hat{n}\):
\[ \hat{v} \cdot \hat{n} = (\hat{i} - \hat{j}) \cdot (x\hat{i} + y\hat{j} + z\hat{k}) = x - y = 0 \]
This gives us:
\[ x - y = 0 \quad \text{(Equation 3)} \]

### Step 5: Solve the equations
From Equations 2 and 3, we have:
1. \(x + y = 0\)
2. \(x - y = 0\)

From Equation 3, we can conclude that \(x = y\). Substituting \(y = -x\) from Equation 2 into this gives:
\[ x = -x \implies 2x = 0 \implies x = 0 \]
Thus, \(y = 0\).

### Step 6: Substitute back to find \(z\)
Substituting \(x = 0\) and \(y = 0\) into Equation 1:
\[ 0^2 + 0^2 + z^2 = 1 \implies z^2 = 1 \implies z = \pm 1 \]
Therefore, we can take:
\[ \hat{n} = \hat{k} \]

### Step 7: Calculate \(|\hat{w} \cdot \hat{n}|\)
Now we need to calculate \(\hat{w} \cdot \hat{n}\):
\[ \hat{w} \cdot \hat{n} = (\hat{i} + 2\hat{j} + 3\hat{k}) \cdot \hat{k} = 0 + 0 + 3 = 3 \]
Thus, the magnitude is:
\[ |\hat{w} \cdot \hat{n}| = |3| = 3 \]

### Final Answer
The value of \(|\hat{w} \cdot \hat{n}|\) is:
\[ \boxed{3} \]